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February 1, 2012February 10, 2012

Logic = Two State Probability

Logic consists of the following objects: 0, 1, AND, OR, and NOT. From these objects, the rest of logic can be built. Note that 0 is sometimes written as false and 1 as true. I will preserve the 0,1 notation since it makes the mappings more obvious.

Logic is two state probability, and will be proven by mapping all of the logical objects onto Probabalistic equivalents, then showing they are identical for all possibilities. All the mappings will be written as B -> P (E) where B is the boolean object, P is the probabalistic equivalent, and E is an optional informal explanation of the probabalistic equivalent. E is provided because of the intuitively appealing nature of some of the mappings:

0 -> 0 (Impossible event [0%])

1 -> 1 (Certain event [100%])

x AND y -> x*y (Probability of x and y happening)

x OR y -> x+y – x*y (Probablity of x or y happening)

NOT x -> 1- x (Probablity of x not happening)

Now for the proof. Recall that the only two values are 0 and 1 so the proof need only hold over those two discrete states:

x AND y = 1 iff x=1 and y=1.

xy = 1 iff x=1 and y=1

OR is more easily done by exhausting all possibilities:

0 OR 0 = 0 -> 0+0 – 0*0 = 0

0 OR 1 = 1 -> 0+1 – 0*1 = 1

1 OR 0 = 0 -> 1+0 – 1*0 = 1

1 OR 1 = 1 -> 1+1 – 1*1 = 1

Ditto for Not:

NOT 0 = 1 -> 1 – 0 = 1

NOT 1 = 0 -> 1 – 1 = 0

Does anyone see any problems with the above?

One way of summarizing the results is to state probability subsumes logic, or that logic is simply probability at the extreme poles, where it reduces to the above.

Thank you for reading! Ultimately, I was just saying that logic could be seen as a subset of probability, that one could do away with logic, replacing it with probability. However, if there was something specific that you found unclear, please let me know and I can try to elaborate.